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Are there known examples of compact infinite dimensional manifolds?

The word "manifold" is important.

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    $\begingroup$ How do you define an infinite dimensional manifold? Is it modelled on the countable product of lines? $\endgroup$ – Igor Belegradek Oct 25 '13 at 0:56
  • $\begingroup$ If this is your definition, then the answer is no, because if $K$ is a compact subset of a manifold $Y$ modelled on the countable product of lines, then $Y$ and $Y-K$ are homeomorphic. $\endgroup$ – Igor Belegradek Oct 25 '13 at 1:04
  • $\begingroup$ just modelled on a vector space. like Banach manifold, Frechet manifold. $\endgroup$ – user8991 Oct 25 '13 at 1:07
  • $\begingroup$ Any separable Frechet (or Banach) space is homeomorphic to the countable product of lines, so my answer above applies. $\endgroup$ – Igor Belegradek Oct 25 '13 at 1:18
  • $\begingroup$ It seems to me Pietro Majer already addressed this in his answer here: mathoverflow.net/a/143737/2926 Right? $\endgroup$ – Todd Trimble Oct 25 '13 at 2:01
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The empty space is a manifold of any dimension.

No, seriously, let's assume that "manifold" means a Hausdorff space in which every point has an open neighborhood homeomorphic to an open subset of a topological vector space. If the manifold is compact and nonempty then the vector space must be locally compact. As far as I know, that makes it finite-dimensional.

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Yes. Compact Hilbert cube manifolds, for instance.

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